Convex Integration Solutions for the Geometrically Non-linear Two-Well Problem with Higher Sobolev Regularity

2019/05/29 by Della Porta, Francesco, Rüland, Angkana
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1905.12521

Abstract

In this article we discuss higher Sobolev regularity of convex integration solutions for the geometrically non-linear two-well problem. More precisely, we construct solutions to the differential inclusion ∇ u∈ K subject to suitable affine boundary conditions for u with K:= SO(2)[1 · amp; δ
0 · amp; 1] ∪ SO(2)[1 · amp; -δ
0 · amp; 1] such that the associated deformation gradients ∇ u enjoy higher Sobolev regularity. This provides the first result in the modelling of phase transformations in shape-memory alloys where K^qc ≠ K^c, and where the energy minimisers constructed by convex integration satisfy higher Sobolev regularity. We show that in spite of additional difficulties arising from the treatment of the non-linear matrix space geometry, it is possible to deal with the geometrically non-linear two-well problem within the framework outlined in \citeRZZ18. Physically, our investigation of convex integration solutions at higher Sobolev regularity is motivated by viewing regularity as a possible selection mechanism of microstructures.

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